A reachability preserver is a basic kind of graph sparsifier, which preserves the reachability relation of an $n$-node directed input graph $G$ among a set of given demand pairs $P$ of size $|P|=p$. We give constructions of sparse reachability preservers in the online setting, where $G$ is given on input, the demand pairs $(s, t) \in P$ arrive one at a time, and we must irrevocably add edges to a preserver $H$ to ensure reachability for the pair $(s, t)$ before we can see the next demand pair. Our main results are: -- There is a construction that guarantees a maximum preserver size of $$|E(H)| \le O\left( n^{0.72}p^{0.56} + n^{0.6}p^{0.7} + n\right).$$ This improves polynomially on the previous online upper bound of $O( \min\{np^{0.5}, n^{0.5}p\}) + n$, implicit in the work of Coppersmith and Elkin [SODA '05]. -- Given a promise that the demand pairs will satisfy $P \subseteq S \times V$ for some vertex set $S$ of size $|S|=:\sigma$, there is a construction that guarantees a maximum preserver size of $$|E(H)| \le O\left( (np\sigma)^{1/2} + n\right).$$ A slightly different construction gives the same result for the setting $P \subseteq V \times S$. This improves polynomially on the previous online upper bound of $O( \sigma n)$ (folklore). All of these constructions are polynomial time, deterministic, and they do not require knowledge of the values of $p, \sigma$, or $S$. Our techniques also give a small polynomial improvement in the current upper bounds for offline reachability preservers, and they extend to a stronger model in which we must commit to a path for all possible reachable pairs in $G$ before any demand pairs have been received. As an application, we improve the competitive ratio for Online Unweighted Directed Steiner Forest to $O(n^{3/5 + \varepsilon})$.

翻译：可达性保持器是一种基本的图稀疏化结构，它能在给定规模为|P|=p的需求对集合P中，保持n节点有向输入图G的可达性关系。本文提出了在线场景下的稀疏可达性保持器构造方法，其中图G作为输入给出，需求对(s, t) ∈ P逐个到达，且必须在看到下一个需求对之前，不可撤销地向保持器H中添加边以确保(s, t)的可达性。我们的主要成果包括：——存在一种构造方法，能保证保持器的最大规模满足$$|E(H)| \le O\left( n^{0.72}p^{0.56} + n^{0.6}p^{0.7} + n\right)$$。该结果在多项式意义上改进了Coppersmith和Elkin [SODA '05]工作中隐含的先前在线上限O( min{np^{0.5}, n^{0.5}p}) + n。——若承诺需求对满足P ⊆ S × V（其中顶点集S的规模|S|=:σ），则存在构造方法能保证保持器最大规模满足$$|E(H)| \le O\left( (npσ)^{1/2} + n\right)$$。对于P ⊆ V × S场景，通过略微不同的构造可获得相同结果。这在多项式意义上改进了先前在线上限O(σn)（常识性结果）。所有构造均具有多项式时间复杂度、确定性特点，且无需预知p、σ或S的数值。我们的技术还使离线可达性保持器的当前上限获得了小幅多项式改进，并可扩展至更严格的模型——在接收任何需求对之前，必须为G中所有可能可达对预先确定路径。作为应用，我们将在线无权有向斯坦纳森林的竞争比改进至O(n^{3/5 + ε})。